OBJETIVOS Y POLÍTICAS DE GESTIÓN DEL RIESGO E INSTRUMENTOS FINANCIEROS

In this section, analysis and simulation describe the performance of the hybrid ap- proach as a communication model in WSNs in the area of k-connectivity and availabil- ity of disjoint paths. The graph theoretic definitions for k-connectivity were briefly described previously.

From the perspective of a communication network and WSNs, there is interest in providing multiple disjoint, mutually independent paths for each node to communicate with the sink or data collection center. The simulation results presented in this section do not consider the existence of multiple paths between all pairs of nodes, but only between each node and a centrally located sink. The motivation for such an approach is the typical traffic model for WSNs being source-to-sink as described earlier.

The network model for the results presented in this section are exactly similar to that in Section a.

The interest of the following analysis is to prove that the hybrid communication model which involves using sensor motes capable of omni-directional and sectorized uni-directional transmission is capable of superior performance when compared to a network using motes that only have omni-directional capability.

Lemma 3 For a random undirected graph of n nodes if edges are added to the empty graph in an order chosen randomly and uniformly from the n₂
! possibilities, then almost surely the graph that results from the edge additions becomes k-connected when it achieves a minimum degree of k. For large n,

P rob(G is k-connected) = P rob(dmin ≥ k) (4.18)

where dmin is the minimum degree (defined in previous sections) per node.

The above has been proved for random graphs in [41] and [40] for graphs with pathloss models.

For the interest of this thesis in WSNs with low node densities, an upper bound for a probability of k-connectivity is computable by considering the probability that the minimum degree of each node in the network graph is greater than or equal to k. In topological terms, this is equivalent to every node in the network having nneigh

neighbors such that nneigh ≥ k. Thus, the probability of dmin ≥ k would give the

upper bound that is needed.

Results for the same exist in [32] in the context of wireless multi-hop networks with nodes capable of omni-directional communication. Following the nearest neigh- bor methods approach employed in that work and using standard graph theoretical results the upper bound can be computed.

Theorem 2 If PHY B(dmin ≥ k) if the probability of the average minimum degree

being greater than k for a network with hybrid-enabled motes and POM N I(dmin ≥ k)

was that for an omni-directional network then, PHY B(dmin ≥ k) ≥ POM N I(dmin ≥ k)

Proof

The minimum degree probability as a function of node density and transmission radius is known from [32].

POM N I(dmin ≥ k) = 1 − k−1 X N =0 (nπr2₎N N ! · e −nπr2 !n (4.19) Here ρ = n, since by definition ρ = _An but in this case A = 1.

The approximation for computing the required bounds for k-connectivity via computing the probability for a minimum degree requirement on each node is ex- pressed below.

P (G is k-connected) ≤ P (dmin ≥ k) (4.20)

As justified earlier, the use of the hybrid approach enables activation of all sectors, thus extending the reach of the sensor mote along all directions. While analytically evaluating this approach, the capability of all sectors to be activated depending on uni-cast traffic awaiting transmission helps extend Eq. (4.19) by substituting for the

transmission radius r with r0 in accordance with the relationship in Eq. (1.3). In the following equations, the minimum degree probability in the omni-directional is denoted by POM N I(dmin ≥ k) and in the hybrid case as PHY B(dmin ≥ k). Eq. (4.19)

can now be re-written as,

PHY B(dmin ≥ k) = 1 − k−1 X N =0 (nπr02)N N ! · e −nπr02 !n (4.21)

Using Eq. (1.3) substituting r0 as r q 2π α so that PHY B(dmin ≥ k) = 1 − k−1 X N =0 (nπ2π_αr2)N N ! · e −nπ2π αr2 !n = 1 − k−1 X N =0 (2nπ_α2r2)N N ! · e −2nπ2r2 α !n = 1 − k−1 X N =0 (2nπ2_r2₎N αN_{N !} · e −2nπ2r2 α !n (4.22)

From Eq. (1.3), Eq. (4.13) and with the expansion in Eq. (4.22) it can be con- cluded that,

PHY B(dmin ≥ k) ≥ POM N I(dmin ≥ k) (4.23)

The hybrid case is equivalent to the omni-directional case when hypothetically, a beamwidth setting of 2π is used. For all other settings, the hybrid case will thus have a higher probability of disjoint paths in the network deployment.

The simulations below explicitly support this claim. The nodes are assumed to be static, with uniform random distribution and capable of both omni-directional and

directional communications. Directional communications is modeled via sectorized uni-directional antennas, dividing the entire omni-directional region of 2π radians into a number of sectors according to the antenna beamwidth. Each sector can be activated, one at a time so that at any instant the node may appear to be equivalent to a uni-directional antenna and that reception is omni-directional. In the omni- directional mode, each node is capable of transmitting at a radius r. When switched to the uni-directional mode, each node is capable of transmitting at a radius r0 related to r by Eq. (1.3), in each sector.

The results shown below are based on a randomly distributed network of nodes in a unit square. There is a centrally located sink at coordinates (0.5, 0.5). The interest of these simulations is in studying the effect of node density, transmission radii and uni-directional antenna beamwidth on the k-connectivity of a randomly deployed network of sensor nodes. The attempt begins by computing the probability of 2-connectivity, or the probability that every node in the network deployment will have at least 2 disjoint mutually independent paths to the centrally located sink. 1000 random topologies were generated to be able to compute the probability. Mutually independent paths are computed using standard disjoint path algorithms, using min- cut/max-flow techniques and link reversals that provide optimal sets of disjoint paths as mentioned in [42] and [43]. To understand the relationship with node density and transmission radius empirically, the normalized r was varied between 0.05 and 0.45 and n, the node density, between 10 and 100. This is basically the probability of 2-connectivity. The effects of varying the beamwidth from π/6 to π/3 was also demonstrated by appropriate configurations for the simulations. These plots are shown below.

Fig. 14 and Fig. 15 describe the probability of 2-connectivity over varying trans- mission radii, node density and antenna beamwidth.

Fig. 14. Probability of Existence of Two Mutually Disjoint Paths for All Nodes in the Network- Varying Transmission Radius r

Fig. 15. Probability of Existence of Two Mutually Disjoint Paths for All Nodes in the Network- Varying Node Density n

It can be seen from the first plot with a constant n and varying r that again, the hybrid approach provides a very substantial non-zero probability even between the lower transmission radii settings of 0.15 and 0.25. At a setting of 0.25, the hybrid approach out performs the omni-directional setting by almost 40%. When the operational transmission radius is set to a high 0.4, the improvement is almost around 80% as can be seen.

The second plot in Fig. 15 describes the effect of varying node density n for a constant r of 0.2. Intuitively with increasing node density, the omni-directional set- ting is able to climb to higher probabilities, as seen for the maximum node density of 100 that is considered for these simulations, the probability for an omni-directional configuration reaches around 0.7. In contrast, the hybrid approach was at a prob- ability of more than 0.7 around a node density of just 40. This emphasizes on the improved performance available when the hybrid approach is employed even at lower node densities. At the interim node density of 50, the hybrid approach out performs an omni-directional only setting by more than 90%.

To further demonstrate the improvements in terms of the availability of disjoint paths for each node, another set of simulations are presented that use the metric Average Number of Disjoint Paths for the Network. This metric represents the average number of paths all member nodes in the network deployment possesses towards the centrally located sink.

Results for varying n and r are presented in Fig. 16 and Fig. 17. For the first plot, a very low node density of n = 10 was considered. 10 nodes distributed over a unit square, is usually a very sparse deployment even for a normalized radius of say, 0.2 for an omni-directional configuration. Interestingly enough, the hybrid setting with r at 0.2, meaning that for α = π/6, r0 is around 0.69, the average number of

Fig. 16. Average Number of Mutually Disjoint Paths for All Nodes in the Network - Varying Transmission Radius r

Fig. 17. Average Number of Mutually Disjoint Paths for All Nodes in the Network - Varying Node Density n

disjoint paths was around 7. For the omni-directional setting, the network was able to even reach 1-connectivity.

For varying n, there is an almost linear relationship in terms of the incremental gains achievable from using the hybrid approach. At the maximum setting of node density 100, the hybrid approach provides around 17, 25 and 40 disjoint paths on an average for the network at the beamwidth settings of π/3, π/4 and π/6 respectively. The omni-directional setting even at the maximum node density of 100 could barely make an average value of around 3 mutually independent paths.